Genus integration, abelianization and prequantization

Given a Lie algebroid, I will explain how  to associate to it a groupoid by using a path-space construction where the underlying homotopies are supported in surfaces with arbitrary genus.  The obstruction to smoothness of this genus integration is controlled by extended monodromy groups. For a general algebroid A, its genus integration is the abelianization of the Weinstein groupoid of A, so this construction can be interpreted as a generalization of the classical Hurewicz theorem. I will illustrate it in the case where A is the central extension of the tangent bundle of a manifold determined by a closed 2-form, where one recovers the prequantization condition and the interpretation of differential cohomology/characters in degree 1 as principal circle bundles with connection. This talk is based on joint work with I Contreras (Amherst College).